On Randomly k-Dimensional Graphs

TL;DR

This paper investigates the properties of randomly k-dimensional graphs, where every k-vertex subset forms a basis, expanding understanding of metric dimensions in graph theory.

Contribution

It introduces the concept of randomly k-dimensional graphs and explores their properties, contributing new insights into metric bases in graph theory.

Findings

Characterization of randomly k-dimensional graphs

Conditions under which a graph is randomly k-dimensional

Properties and examples of such graphs

Abstract

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ with minimum cardinality is called a basis of $G$ and its cardinality is the metric dimension of $G$. A connected graph $G$ is called randomly $k$-dimensional graph if each $k$-set of vertices of $G$ is a basis of $G$. In this paper, we study randomly $k$-dimensional graphs and provide some properties of these graphs.